On the standard conjecture for a fourfold with $1$-parameter fibration by Abelian varieties

Abstract

It is proved that the Grothendieck standard conjecture $B(X)$ of Lefschetz type holds for a smooth complex projective 4-dimensional variety $X$ provided that there exists a morphism of $X$ onto a smooth projective curve whose generic scheme fibre is an Abelian variety with bad semi-stable reduction at some place of the curve.